On a characteristic problem for a loaded hyperbolic equation

Abstract

The paper studies a loaded hyperbolic equation with one-dimensional wave equation in its main part. In the loaded components there are two points, which are distributed respectively along a pair of intersecting characteristics at a constant speed. For such an equation we study the Cauchy problem with characteristics of the one dimensional wave equation belonging to any of the pair of intersecting straight lines. If to impose certain conditions at the boundary points on function presenting Cauchy data and the derivatives of the first, second and third orders we can prove the existence and uniqueness of the problem. The proof of the existence and uniqueness of the solution follows directly from the method of its production. We consider also issues concerning domains of dependence, influence, and definitions for Cauchy data that are specified on one of the characteristic curves. The above verifies once more the thesis of load influencing on posing of various initial - boundary value problem for partial differential equations.